It sounds to me like your daughter could really benefit from being in a gifted school or a gifted classroom. The social benefit of being around other like-minded kids is huge. That's one of the main reasons I wanted to make sure my daughter was in a gifted classroom. If a gifted school or class is not an option where you live, skipping a grade can help. It's better for kids to be with intellectually-matched peers than with age-matched peers. It is also very typical for gifted kids who are bored and not appropriately challenged to have ADHD-like behavior. Good luck with testing and figuring everything out!